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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>

<b>thus </b><a NAME="E1:10"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R2" target="_self">well_orders</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> is <a href="wellord1.html#V2" target="_self">well-ordering</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R1" target="_self">is_well_founded_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D5" target="_self">WELLORD1:def 5</a><br/>

<b>hence </b><a NAME="E2:10_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V1">reflexive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V8">transitive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V4">antisymmetric</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V6">connected</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="wellord1.html#V1" target="_self">well_founded</a> )
 <b>by </b><i><a class="ref" href="wellord1.html#T5" target="_self">Th5</a>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a>, <a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D4" target="_self">WELLORD1:def 4</a><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:10"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V1">reflexive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V8">transitive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V4">antisymmetric</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V6">connected</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="wellord1.html#V1" target="_self">well_founded</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D4" target="_self">WELLORD1:def 4</a><br/>

<b>hence </b><a NAME="E3:10"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R1" target="_self">is_well_founded_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="wellord1.html#T5" target="_self">Th5</a>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a>, <a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord1.html#D5" target="_self">WELLORD1:def 5</a><br/>


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